Geometrical Interpretation of Neutrino Oscillation with decay

The geometrical representation of two-flavor neutrino oscillation represents the neutrino's flavor eigenstate as a magnetic moment-like vector that evolves around a magnetic field-like vector that depicts the Hamiltonian of the system. In the present work, we demonstrate the geometrical interpretation of neutrino in a vacuum in the presence of decay, which transforms this circular trajectory of neutrino into a helical track that effectively makes the neutrino system mimic a classical damped driven oscillator. We show that in the absence of the phase factor $\xi$ in the decay Hamiltonian, the neutrino exactly behaves like the system of nuclear magnetic resonance(NMR); however, the inclusion of the phase part introduces a $CP$ violation, which makes the system deviate from NMR. Finally, we make a qualitative discussion on under-damped, critically-damped, and over-damped scenarios geometrically by three different diagrams. In the end, we make a comparative study of geometrical picturization in vacuum, matter, and decay, which extrapolates the understanding of the geometrical representation of neutrino oscillation in a more straightforward way.